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The mechanical response of cellular materials with spinodal topologies

Meng-Ting Hsieh, Bianca Endo, Yunfei Zhang, Jens Bauer, Lorenzo Valdevit*

Mechanical and Aerospace Engineering Department, University of California, Irvine, CA 92697

Abstract

The mechanical response of cellular materials with spinodal topologies is numerically and

experimentally investigated. Spinodal microstructures are generated by the numerical solution of

the Cahn-Hilliard equation. Two different topologies are investigated: ‘solid models,’ where one

of the two phases is modeled as a solid material and the remaining volume is void space; and ‘shell

models,’ where the interface between the two phases is assumed to be a solid shell, with the rest

of the volume modeled as void space. In both cases, a wide range of relative densities and spinodal

characteristic feature sizes are investigated. The topology and morphology of all the numerically

generated models are carefully characterized to extract key geometrical features and ensure that

the distribution of curvatures and the aging law are consistent with the physics of spinodal

decomposition. Finite element meshes are generated for each model, and the uniaxial compressive

stiffness and strength are extracted. We show that while solid spinodal models in the density range

of 30-70% are relatively inefficient (i.e., their strength and stiffness exhibit a high-power scaling

with relative density), shell spinodal models in the density range of 0.01-1% are exceptionally stiff

and strong. Spinodal shell materials are also shown to be remarkably imperfection insensitive.

These findings are verified experimentally by in-situ uniaxial compression of polymeric samples

* Corresponding Author. E-mail: Valdevit@uci.edu

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printed at the microscale by Direct Laser Writing (DLW). At low relative densities, the strength

and stiffness of shell spinodal models outperform those of most lattice materials and approach

theoretical bounds for isotropic cellular materials. Most importantly, these materials can be

produced by self-assembly techniques over a range of length scales, providing unique scalability.

Keywords: Cellular Materials; Mechanical Metamaterials; Spinodal Decomposition; Finite

Element Analysis; Nanofabrication.

1. Introduction

Cellular materials are porous materials consisting of one or more solid phases and void space. For

all cellular materials, Young’s modulus, E and yield (or buckling) strength, 𝜎𝑦 , are a strong

function of the relative density, ρ̅ (i.e., the volume fraction of the solid phase), generally

following a power-law behavior: 𝐸~𝐸𝑠𝜌 ̅𝑛 and 𝜎𝑦~𝜎𝑦𝑠�̅� 𝑚, with 𝐸𝑠 and 𝜎𝑦𝑠 the Young’s modulus

and yield (or buckling) strength of the base material, respectively. The values of the exponents n

and m have a dramatic effect on the mechanical efficiency of the material and are strongly affected

by the topology of the unit cell architecture (Fleck et al., 2010).

The mechanical properties of open-cell, strut-based cellular materials like foams and lattices have

been studied extensively (Ashby et al., 2000; Deshpande et al., 2001b; Gibson and Ashby, 1997).

Depending on the coordination number (number of bars meeting at each node) and the balance

between the number of states of self-stress and internal mechanisms, beam-based topologies can

be either bending-dominated or stretching-dominated: in the former, local deformation upon global

loading (e.g., compression) occurs primarily by bending of the struts (a weak deformation mode);

by contrast, in the latter the vast majority of the strain energy is stored by axial deformation of the

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struts (Deshpande et al., 2001a). The implication is that stretching-dominated lattices are much

more mechanically efficient than bending dominated lattices, with effective stiffness and yield

strength scaling as 𝐸, 𝜎𝑦 ~ (ρ̅)1 (versus 𝐸 ~ (ρ̅)2 and 𝜎𝑦 ~ (ρ̅)1.5 for bending-dominated 3D

lattices and stochastic foams) (Fleck, 2004). In all cases, at low relative density, strut buckling

limits the strength of both stretching- and bending-dominated lattices. For nearly any strut-based

lattice architecture, the effective elastic buckling strength scales with ρ̅ as 𝜎𝑒𝑙 ~ (ρ̅)2 (Ashby,

2006, 1983; Deshpande et al., 2001b), dramatically reducing mechanical performance at low

relative densities. For most solid-strut topologies, the yielding-to-buckling transition occurs at a

relative density of the order of ~1% (Gibson and Ashby, 1988).

Hollow-strut lattices can potentially delay the onset of buckling, shifting the yielding-to-buckling

transition to ρ̅<0.1% (Bauer et al., 2017; Meza et al., 2014; Pingle et al., 2011; Schaedler et al.,

2011; Torrents et al., 2012; Valdevit et al., 2013, 2011). However, their mechanical properties are

often much worse than analytical predictions suggest: for example 𝐸 ~ (ρ̅)1.61 and 𝜎𝑦 ~ (ρ̅)1.76

has been experimentally extracted for hollow-strut ceramic octet lattices (Meza et al., 2014). The

hollow nodes of these lattices were shown to deform through bending (regardless of the

coordination number); this phenomenon, coupled with stress concentration at the nodes, causes a

knock-down from classical scaling laws, which do not account for nodal topology (Bauer et al.,

2016; Bauer et al., 2017; Meza et al., 2014; Portela et al., 2018; Torrents et al., 2012; Valdevit et

al., 2013). Local node buckling (at low relative densities) and high sensitivity to imperfection

(waviness of the strut and non-ideal strut cross section) during fabrication of these structures also

contribute to the reduction of strength (Salari-Sharif et al., 2018; Valdevit et al., 2013) . Adding

fillets to the nodes efficiently reduces these stress concentrations, enhancing the strength of

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hollow-strut lattices. Increasing node-smoothening gradually converts hollow-strut lattices to

shell-based cellular materials (Han et al., 2015).

Shell-based cellular materials can be classified in closed-cell and open-cell materials, based on the

connectivity of their porosity. The former include closed-cell stochastic foams (Gibson and Ashby,

1988) and 3D periodic arrangements of polygonal cells, such as closed-cell octet and cubic-octet

regular foams (Berger et al., 2017) and quasi closed-cell inverse opals (do Rosário et al., 2017).

While some of these materials are remarkably efficient (e.g., the nearly-isotropic cubic-octet

regular foam approaches the theoretical Hashin-Shtrikman stiffness bounds (Berger et al., 2017),

their topology still results in stress localization near face edges and vertices, and hence sub-optimal

material utilization.

By contrast, open-cell shell-based materials can be constructed as a thin, smooth, interconnected

membrane that occupies a 3D volume, with interconnected porosity. These topologies usually take

the form of a locally area-minimized geometry (Meeks and Perez, 2011), and are characterized by

a mean curvature approximately equal to zero everywhere on the surface. The most famous

examples are triply periodic minimal surfaces (TPMS), such as the Schwarz P surface (Hyde et

al., 1996; Schwarz, 1890), the Schwarz D surface (Hyde et al., 1996; Schwarz, 1890) and the

gyroid surface (Schoen, 1970). The uniform curvature of these surfaces has been numerically

shown to dramatically reduce local stress intensification upon macroscopic loading (Rajagopalan

and Robb, 2006). Furthermore, Schwarz P shell and Schwarz D shell materials were shown to be

stretching-dominated, with E and 𝜎𝑦 on par with or better than most of the existing lattice

architectures (Han et al., 2017, 2015; Nguyen et al., 2017). A similar class of open-cell thin shell

materials was derived from simple-cubic (SC), face-centered cubic (FCC), and body-centered

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cubic (BCC) tube-lattices. These topologies have tailorable anisotropy and can achieve elastic

moduli higher than the hollow truss lattices they are derived from (Bonatti and Mohr, 2018).

Unfortunately, all these thin shell materials have two shortcomings: (i) they are significantly

imperfection sensitive, as any surface roughness or waviness due to manufacturing imperfections

would cause them to buckle prematurely under compression (Han et al., 2017; Hutchinson and

Thompson, 2018); and (ii) they are generally difficult to fabricate with scalable manufacturing

processes (inverse opal and gyroid materials can be fabricated at the nanoscale, but upscaling is

challenging). The development of architected materials that can be fabricated at macroscale

dimensions while still retaining dimensional control of their nano/micro-scale features is a very

active research endeavor. The driving force has been the quest for scale-up of the well-known

beneficial size effects on strength that metallic and ceramic materials exhibit when one or more

component dimensions are reduced to the nanoscale. Excellent performance has been

demonstrated in solid ceramic nanolattices (Bauer et al., 2016), hollow metallic (Schaedler et al.,

2011; Zheng et al., 2016) and ceramic (Meza et al., 2014) micro and nano-lattices and nano-shell-

based cellular materials (Khaderi et al., 2017, 2014). The practical applicability of all these nano-

architected materials, though, is dramatically hindered by scalability challenges that derive from

their manufacturing process (Bauer et al., 2017; Bishop-Moser et al., 2018) . Approaches based on

self-assembly, possibly combined with additive manufacturing at a larger scale, have the potential

to dramatically increase scalability.

Spinodal decomposition is a near-instantaneous diffusion-driven phase transformation that

converts a single-phase material into a two-phase material (one possibly being void space), with

the two phases arranged in a bi-continuous topology and separated by a surface with nearly

uniform negative Gaussian curvature and nearly zero mean curvature (Cahn, 1961; Jinnai et al.,

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1997). Spinodal decomposition may occur by a variety of physical processes, e.g., the dealloying

of an Au/Ag solid solution (Hodge et al., 2007) or the heating/cooling of an emulsion gel through

a critical point (Lee and Mohraz, 2010). The former results in a gold solid spinodal structure with

~50% porosity and nanoscale feature size (Fig. 1a), whereas the latter gives rise to a ~50%/50%

mixture of two liquids with microscale feature size, called a bicontinuous interfacially jammed

emulsion gel (bijels) (Fig. 1b). While thermodynamics favors the growth of the feature size over

time, this process can be arrested (thereby controlling the length scale), by lowering the

temperature (in the case of nano-porous metal) or by jamming the interface with solid nano-

particles (in the case of bijels). Bijels can be used as templates for the development of structural

and functional micro-architected materials, via a suite of materials conversion processes (Lee et

al., 2013; Lee and Mohraz, 2011). For example, one of the liquid phases can be replaced by a

photosensitive monomer, which is subsequently photopolymerized, and the other liquid phase

removed, resulting in a ~50% dense polymeric cellular spinodal architecture. The surface of this

cellular solid can be subsequently coated with metal or ceramic, and the polymer ultimately

dissolved, with the end result being a stochastic shell-based architected material with spinodal

topology (Lee and Mohraz, 2010) (Fig. 1c). This process is extremely scalable, and can result in

rapid fabrication of macro-scale shell-based cellular materials with microscale feature size. The

stochastic nature of this shell-based architecture, though, raises questions about its mechanical

efficiency.

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Figure 1 – Examples of self-assembled spinodal topologies at different length scales: (a) nano-

porous Au, obtained by selective etching of Au/Ag solid solutions (reproduced from (Hodge et al.,

2007)); (b) spinodally decomposed bijel (reproduced from (Lee and Mohraz, 2010)); (c) micro-

porous metallic shell spinodal architected material, obtained by material conversion of spinodally

decomposed bijels (reproduced from (Lee and Mohraz, 2010)).

In this work, we thoroughly investigate the mechanical response of both solid and shell-based

architected materials with spinodal topologies. We numerically generate 3D spinodal cellular

materials with different relative densities and characteristic feature sizes and extract shell-based

spinodal cellular materials from the interface between the solid and the void space. We compute

stiffness and strength under uniaxial compression by finite element simulations, which are verified

by mechanical experiments conducted on polymeric samples produced by 2-photon

polymerization Direct Laser Writing. We show that while solid spinodal cellular materials scale

poorly with relative densities, shell spinodal architected materials are remarkably efficient, with

effective Young’s modulus and yield strength scaling with the relative density as �̅� 1.2-1.4.

Importantly, spinodal shell topologies are shown to be largely imperfection insensitive, and hence

robust against elastic buckling failure modes at ultra-low densities. These findings reveal that shell

spinodal architected materials perform better than many available periodic topologies (both strut

and shell-based), particularly at ultra-low density. These mechanical characteristics, combined

with the ability to self-assemble at multiple length scales, make shell-based spinodal architected

materials a remarkable class of cellular solids.

2. Numerical construction of spinodal topologies

2.1 Generation of solid spinodal topologies

Spinodal structures are formed through spinodal decomposition, a diffusion-type phase separation

process that separates a compound into a mechanical mixture of phase A and phase B solid

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solutions by reducing the total free energy of the system (Porter et al., 1981). The phase separation

processes during spinodal decomposition can numerically be described with phase field

approaches (Biener et al., 2009; Crowson et al., 2009, 2007). One of the most commonly used

formulations is the Cahn-Hilliard equation (Cahn and Hilliard, 1958; Sun et al., 2013), which can

be written as:

𝜕𝑢

𝜕𝑡= ∆[

𝑑𝑓(𝑢)

𝑑𝑢− 𝜃2∆𝑢] (1)

where 𝑢(𝑥, 𝑦, 𝑧, 𝑡), is the concentration difference between the two phases (phase A and phase B)

at a coordinate (x,y,z) at the evolution time, t (𝑢(𝑥, 𝑦, 𝑧, 𝑡) is bounded between -1 and 1, with 𝑢 =

−1 denoting full phase A and 𝑢 = 1 denoting full phase B), and ∆ is the Laplacian operator. At

time 𝑡 = 0, the system is nearly homogeneous, with 𝑢~0 everywhere. As the simulation starts,

phase separation is driven by a free-energy function with a double well, which can be chosen as

𝑓(𝑢) = 1

4(𝑢2 − 1)2. Regions of phase A and phase B quickly develop, with 𝜃 denoting the width

of the interface between the two phases (here set to 1.1, as suggested by (Sun et al., 2013)). As

time progresses the size of the single-region domains increases and the curvature of the interface

decreases. For a cellular material, one of the two phases represents void space. This equation is

discretized by the Finite Difference method, and solved on a cubic domain of 100 X 100 X 100

nodes, subject to periodic boundary conditions. The solution procedure is nearly identical to that

reported in (Sun et al., 2013); a synopsis is reported in Appendix 1 for completeness. Solid spinodal

volumes at 5 different evolution times, 𝑡1 − 𝑡5 (and hence different domain sizes and surface

curvatures) and 4 different relative densities, �̅�1 = 20%, �̅�2 = 30%, �̅�3 = 50% and �̅�4 = 70%,

are extracted for post-processing (Fig. 2). Spinodal volumes were subsequently sliced in 2D

images and the stack of images was imported in the commercial software Simpleware ScanIP to

generate a 3D tetrahedral finite element mesh for subsequent mechanical characterization.

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Figure 2 – 3D spinodal solid models with different relative densities, �̅� = 0.2, 0.3, 0.5, 𝑎𝑛𝑑 0.7, extracted at different evolution times, t1, t3, and t5.

2.2 Calculation of the characteristic feature size

Whereas periodic architected materials have a very well-defined unit cell size, for stochastic

spinodal solid this definition is fuzzier. Nonetheless, a measure of the characteristic feature size,

𝜆, is necessary to enable comparison with periodic cellular materials. This dimension can be

defined as the inverse of the dominant frequency in a Fast Fourier Transform (FFT) of the spinodal

model (Jinnai, 2004; Kwon et al., 2010). In order to graphically illustrate the relationships between

real domain and frequency domain, FFTs are initially extracted on two-dimensional slices of the

solid spinodal topologies (Fig. 3). The results show three important features: (i) the intensity

profiles in the FFT images are all nearly axisymmetric, indicating that all our spinodal models are

isotropic; (ii) the dominant frequencies shrink as the evolution time increases, indicating that the

characteristic feature size increases with evolution time; (iii) the dominant feature size is

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unaffected by the relative density of the solid spinodal models and only depends on evolution time,

suggesting that this unique characteristic length scale can be used to uniquely describe the

morphology evolution of the system (in lieu of time), a description that we adopt henceforth for

both solid and shell spinodal topologies.

Figure 3 – 2D slices of spinodal solid models in (a) space domain and (b) frequency domain, at

different relative densities (�̅� = 0.2, 0.3, 0.5, 𝑎𝑛𝑑 0.7) and different evolution times (t1, t3, and t5).

The characteristic feature size is extracted from the FFT transforms of the models, and depends

on the evolution time, but not on the relative density.

To properly measure the characteristic feature size for each topology, FFTs are extracted for full

three-dimensional spinodal models. Plots of amplitude VS frequency along the x-direction are

extracted and depicted in Fig. 4a (the specific direction is immaterial, as the models were shown

to be isotropic). Notice that the frequency has been multiplied by the domain size L in Fig. 4a;

with this normalization, the x-position of the dominant frequency peak represents the number of

unit cells in the domain, 𝑁 = 𝐿/𝜆, with 𝜆 the characteristic feature size.

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Fig. 4a clearly shows that the characteristic feature size increases with evolution time, as

qualitatively observed in the 2D FFTs. This time evolution is explicitly plotted in Fig. 4b, where

the characteristic feature size is shown to scale with the cubic root of the evolution time, as

expected by the LSW (Lifshitz-Slyozov-Wagner) theory of spinodal decomposition (Baldan,

2002; Kwon et al., 2010), for all but the first simulation. The slight disagreement on the first

simulation point is attributed to an incomplete phase separation in the Cahn-Hillard evolution

model (i.e., evolution is not yet fully completed at that short time, and feature growth had not fully

started). This finding confirms that the numerical topologies obtained with the technique described

in sec. 2.1 possess the characteristic features of naturally occurring spinodal structures.

Figure 4 – Time evolution of the inverse of the characteristic feature size, as extracted from 3D

FFT analyses of solid spinodal models. (a) FFT amplitude along the x direction at five evolution

times; the frequency with highest amplitude is the inverse of the characteristic feature size. (b)

The characteristic feature size scales with the cubic root of the evolution time (normalized by t5),

as predicted by the LSW theory of spinodal decomposition.

2.3 Generation of shell spinodal topologies

Spinodal shell geometries are generated by extracting the interface between solid and void phases

of spinodal solid models with a three-step process: (i) the stack of 2D images for each model is

imported in Matlab and a 3D matrix containing the phase information of any point in the cubic

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domain at a particular instant in time is generated; (ii) the Matlab ‘isosurface’ function is used to

extract the interface between solid and void phases; (iii) the Matlab ‘surface smoothening’

algorithm is applied the isosurface patches, in order to remove sharp geometric transitions. The

resulting three-dimensional surface is then meshed with triangular shell elements within Matlab

and subsequently imported in the commercial finite element software Abaqus for further

mechanical analysis. The relative density of spinodal shell topologies is tuned by controlling the

shell thickness in Abaqus. With this procedure, we generate spinodal shell topologies from each

of the 5 solid spinodal topologies with 50% relative density discussed above (each with a different

evolution time, and hence domain size and surface curvature); three such topologies are depicted

in Fig. 5a. Once the surface area per volume ratio, S/V, of each model is extracted, the relative

density of the shell spinodal model is calculated as �̅� = 𝑏𝑆/𝑉, with b the shell thickness. This

relationship is plotted in Fig. 5b; notice that the shell thickness has been normalized by the

characteristic domain size, 𝜆, defined in sec. 2.2 above. Using this approach, we generate shell

spinodal structures with relative densities, �̅� = 0.01%, 0.1%, 1%, 5%, 7%, and 10% for further

analysis.

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Figure 5 – (a) 3D spinodal shell models corresponding to the interfaces of spinodal solid models

of �̅� = 0.5, extracted at three different evolution times (t1, t3, and t5). (b) Relative density of shell

spinodal cellular materials, as a function of the shell thickness, b, normalized by the characteristic

feature size, 𝜆. L denotes the cubic domain size.

2.4 Calculation of the surface curvature

While clearly stochastic in nature, spinodal structures possess a remarkable uniform distribution

of surface curvatures, with every surface patch characterized by a near-zero mean curvature and

negative Gaussian curvature. Mean and Gaussian curvatures can be expressed in terms of the

principal curvatures as 𝐻 = 1

2(𝜅1 + 𝜅2) and 𝐾 = 𝜅1 ∙ 𝜅2. To confirm that the models generated in

sec. 2.1 conform to this important signature of spinodal structures, we extract the principal

curvatures at any location in each shell spinodal model with the algorithm described in Appendix

2. The results are plotted in Fig. 6. A number of important results clearly emerge: (i) for all

topologies derived from 50% dense solid models (i.e., same amount of solid and void phases),

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𝐻~0 and 𝐾 < 0, with most surface points within a very narrow range, as predicted by the theory

of spinodal decomposition (Kwon et al., 2010); henceforth, we will use these models to construct

shell spinodal topologies for mechanical characterization; (ii) for topologies derived from solid

models with densities lower (higher) than 50%, 𝐻 < (>)0 and 𝐾 < 0, with most surface points

within a very narrow range; (iii) for all topologies, the magnitudes of the principal curvatures

decrease with evolution time, remaining consistent with the trends mentioned above; this is clearly

consistent with the characteristic feature size coarsening over time, as noticed above.

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Figure 6 – Principal curvature maps of the surface of shell spinodal models, derived from different

relative densities (�̅� = 20%, 30%, 50%, 𝑎𝑛𝑑 70%) and different characteristic features sizes ( 𝜆

𝐿=

1

8,

1

5, 𝑎𝑛𝑑

1

3 ) of solid spinodal preforms. Notice that only the 50%-derived shell topology has

near-zero mean curvature, 𝐻 = 1

2(𝜅1 + 𝜅2), throughout the surface.

3. Mechanical performance of spinodal topologies

3.1 Numerical modeling of mechanical response

The finite element (FE) meshes of the solid and shell spinodal models, generated as described in

sec. 2, are imported in the commercial FE software package Abaqus Standard, and static

simulations are performed to extract the mechanical response of these cellular materials under

uniaxial loading, specifically the effective Young’s modulus, E, and the effective yield strength,

𝜎𝑦. The base material is modeled as isotropic elastic-perfectly plastic, with Young’s modulus,

𝐸𝑠 = 210 𝐺𝑃𝑎 and yield strength, 𝜎𝑦𝑠 = 235 𝑀𝑃𝑎 (the actual values are immaterial, as the

effective properties are normalized by these values). Geometric nonlinearity is allowed during

deformation.

To simulate the uniaxial response of an infinite solid, the following boundary conditions are

applied on the cubic domains (Fig. 7): (i) the nodes on the –x face are restrained from moving in

the x-direction but allowed to move freely in the y- and z-direction; (ii) a compressive

displacement, 𝛿𝑥 = −6, corresponding to a lattice level strain 휀𝑥 = −6%, is applied to all the

nodes on the +x face, which are otherwise free to move in the y- or the z-direction; (iii) all the

nodes on the +y face are free to move in all directions, although their y-displacement is coupled to

that of a reference node (and hence is the same for all nodes on the face); similarly, all the nodes

on the +z face are free to move in all directions, although their z-displacement is coupled to that

of a reference node (and hence is the same for all nodes on the face); (iv) the nodes on the –y face

are constrained from moving in the y-direction but are free to move in the x- or z-direction, and

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the nodes on the –z face are constrained from moving in z- direction but are free to move the in x-

or y- direction. Although not as rigorous as periodic boundary conditions, these boundary

conditions simulate the mechanical response of infinite spinodal structures, ensuring that all plane

faces remain plane and allowing for Poisson’s contraction upon uniaxial compression.

The reaction force (𝐹𝑅 ) integrated on all the nodes on the –x face is extracted through the

simulation. The stress-strain curve for the cellular material can then be generated, with stress and

strain defined as 𝜎 = 𝐹𝑅/𝐿2 and 휀 =𝛿𝑥

𝐿, respectively, with L the cubic domain size. The effective

Young’s modulus of the cellular material, 𝐸, is calculated as the slope of the linear region in the

stress-strain curve, and the effective yield strength, 𝜎𝑦, is extracted as the 0.2% offset strength.

Figure 7 – Finite element model of (a) a solid spinodal model ( �̅� = 50% ) and (b) the

corresponding shell spinodal model. The nodes on the positive x surface are highlighted in red

indicating how the boundary conditions are applied to the models. (c-d) Boundary conditions

applied to a spinodal solid model in the x-y and x-z planes. Red, green, and blue triangles indicate

that translations are constrained along the x, y, and z directions, respectively. The sides opposite

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to those indicated with red, green, and blue triangles are allowed to displace in the x, y, and z

directions, respectively, but with the condition that all nodes on the side displace by the same

amount. The same boundary conditions are applied to spinodal shell models.

3.2 Imperfection sensitivity analysis

Thin shells (and by extension thin shell-based architected materials) are known to be stiff and

strong but significantly sensitive to geometric imperfections (Hutchinson and Thompson, 2018), a

characteristic that significantly limits their load carrying potential in practical applications. To

quantify this effect in spinodal shell models, an imperfection sensitivity analysis is performed.

First, linear eigenvalue (elastic buckling) analyses are performed on the original meshes and the

first ten eigenmodes with positive eigenvalues are extracted from each mesh. Second,

imperfections are introduced in the original meshes as linear combinations of the first ten

eigenmodes with equal weights; the magnitude of the imperfection parameter, 𝜓 = ℎ𝑚𝑎𝑥/𝑏, with

ℎ𝑚𝑎𝑥 the largest displacement from the original mesh and b the shell thickness, is systematically

increased in order to probe the imperfection sensitivity. Finally, post-buckling non-linear quasi-

static Ricks analyses are performed in uniaxial compression, subject to same boundary conditions

described above, in order to capture the deformation evolution in imperfect meshes. Figure 8 and

9 show meshes and stress-strain curves for two different 10% dense shell spinodal topologies,

differing by the size of the characteristic feature, 𝜆, relative to the domain size ((a) 𝜆/𝐿 = 1/8, (b)

𝜆/𝐿 = 1/3). For each topology, the response of the perfect structure (𝜓 = 0) is compared to the

responses of the same topology with increasing amounts of imperfection. A very notable

conclusion emerges: the topology with 8 X 8 X 8 ‘unit cells’ in the domain (Fig. 9a) is absolutely

imperfection insensitive, with imperfection magnitudes as large as 20 times the shell thickness

showing no effect on the mechanical response. Even for very coarse topologies, with only 3 X 3

X 3 ‘unit cells’ in the domain (Fig. 9b), the imperfection sensitivity becomes appreciable only

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when 𝜓~10. (To further emphasize the imperfection insensitivity of these topologies, notice that

the topology with characteristic feature size, 𝜆

𝐿= 1/3 (Fig. 8c-d and Fig. 9b) has much smaller

surface area per unit volume than the topology with 𝜆

𝐿= 1/8 (Fig. 8a-b and Fig. 9a), and hence

larger shell thickness; hence, in absolute terms, the imperfection magnitudes displayed in Fig. 9b

are much larger than those in Fig. 9a (as evident from comparing Fig. 8a-b with Fig. 8c-d),

explaining the emergence of some degree of imperfection sensitivity). Overall, the remarkable

conclusion is that, unlike regular shell-based architected materials, spinodal shell based architected

materials are remarkably imperfection insensitive; this important characteristic is attributed to the

stochastic nature of the spinodal shell topologies. With this conclusion, all subsequent finite

element analyses are performed on perfect meshed, without loss of conservatism.

Figure 8 – Comparisons of perfect (a,c) and perturbed (c,d) meshes for spinodal shell topologies

with characteristic feature size, 𝜆

𝐿=

1

8 (a,b) and

𝜆

𝐿=

1

3 (c,d).The magnitude of the imperfection is

equal to 𝜓 = 20 (b) and 𝜓 = 9.3 (d), respectively. Red circles in (a,b) represent examples of

subtle imperfections.

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Figure 9 – Imperfection insensitivity of shell spinodal models. The non-dimensional imperfection

magnitude, 𝜓, is defined as the largest displacement from a perfect mesh normalized by the shell

thickness. Comparison of post-buckling stress-strain curves of 10% dense spinodal shells with

characteristic feature size, 𝜆

𝐿=

1

8 (a) and for

𝜆

𝐿=

1

3 (b), with three different magnitudes of

imperfection. The constituent material has Young’s modulus, 𝐸𝑠 = 210 𝐺𝑃𝑎 and yield

strength, 𝜎𝑦𝑠 = 235 𝑀𝑃𝑎.

3.3 Experimental verification

The accuracy of the finite element simulations is verified with a selected set of uniaxial

compression experiments. Solid and shell spinodal samples were fabricated at the micro-scale via

two-photon polymerization Direct Laser Writing (DLW), using a Nanoscribe Photonic

Professional GT and a negative tone photoresist (IP-Dip, produced by Nanoscribe GmbH). All

samples were fabricated using the Galvo mode with the 63X, N.A. 1.4 objective lens. Printing

parameters were 35 mW and 17,000 µm/s for laser power and scan speed, respectively. Cubic

samples with edge length, 𝐿 = 140 𝜇𝑚 , are generated, using the solid and shell spinodal

geometries calculated in Sec. 2. A characteristic feature size, 𝜆/𝐿 = 1/5, is used for all samples.

Solid spinodal samples with relative densities of 20%, 30% and 50%, and shell spinodal samples

with relative densities of 5%, 7% and 10% were fabricated, respectively (all shell spinodal

topologies were extracted from a 50% dense solid spinodal, as explained in Sec. 2). After DLW,

samples were first immersed in propylene glycol monomethyl ether acetate (PGMEA) for 20

20

minutes to dissolve the remaining liquid resin, and subsequently in isopropanol for 2 minutes for

final cleaning. See Fig. S1 for SEM images of all samples.

The mechanical response of all spinodal structures was characterized by displacement-controlled

uniaxial compression tests with a maximum strain of 15%, performed at a constant strain rate

of 0.005/𝑠. All tests were performed with an Alemnis Nanoindenter. Load-displacement curves

were recorded and extracted, and converted to stress-strain curves as explained in Sec. 3.1. To

compare experimental results with numerical predictions, finite element models of the same

topologies were generated and loaded in uniaxial compression, as explained in Sec. 3.1. In order

to best approximate the experimental conditions, FE simulations are performed with the bottom

face fully constrained, the lateral faces free and the top face subjected to uniform vertical

displacement but unconstrained laterally. In order to extract the base material properties for the

numerical analyses, a number of IP-Dip cylinders with diameters between 12.5 and 50μm and

aspect ratios of four were DLW printed with the same parameters as the spinodal samples and

tested in uniaxial compression (see Supplementary Material Fig. S2) (ASTM D695-15, 2008). As

the mechanical properties of this DLW acrylate resin are affected by the printing conditions and

feature size (which cannot be kept identical for all the structures being printed), extracting an

accurate stress-strain curve for the base material is challenging. To simplify the model, and in

agreement with the results in Fig. S2, the base material was modeled as elastic-perfectly plastic,

with Young’s modulus, 𝐸𝑠 = 2 𝐺𝑃𝑎, and yield strength 𝜎𝑦𝑠 = 60 𝑀𝑃𝑎. While this simplification

will slightly under-predict the ultimate strength of the structures, it is appropriate for the scope of

this section.

Experimental and numerical stress-strain curves are compared in Fig. 10a-b, for solid and shell

spinodal models, respectively. With the caveat on ultimate strength mentioned above, the

21

agreement is very good throughout, thus validating all the finite element results presented in this

work.

Figure 10 – Comparison of numerical and experimental stress-strain curves for (a) solid and (b)

shell spinodal topologies at different relative densities, loaded in uniaxial compression. The

characteristic feature size, 𝜆

𝐿=

1

5 is used for all models. The insets depict SEM images of the

samples, fabricated by two-photon polymerization Direct Laser Writing in IP-Dip, with a

Nanoscribe GT Photonics Professional, and tested with an Alemnis Nanoindenter. Higher

resolution SEM images of all samples are shown in Fig. S1. The properties of IP-Dip were

measured by micropillar compression tests, and input in the simulations as an elastic-perfectly

plastic solid with 𝐸𝑠 = 2 𝐺𝑃𝑎, and yield strength 𝜎𝑦𝑠 = 60 𝑀𝑃𝑎.

3.4 Scaling laws and mechanical performance

Effective Young’s modulus and yield strength were extracted from finite element calculations

(performed as described in Sec. 3.1) for solid and shell spinodal models, for a range of relative

densities, �̅� and characteristic feature size, 𝜆/𝐿. In all simulations the constituent material was

modeled as isotropic elastic-perfectly plastic, with Young’s modulus, 𝐸𝑠 = 210 𝐺𝑃𝑎 and yield

strength, 𝜎𝑦𝑠 = 235 𝑀𝑃𝑎. These values are representative of a relatively low yield strain metal,

22

with yield strain 휀𝑦~0.001. Most results presented here are general and independent on the choice

of base material.

The effective Young’s modulus and yield strength of solid spinodal models depending on ρ̅ are

depicted in Fig. 11. Notice that 𝐸 ~ (ρ̅)2.0−2.6 and 𝜎𝑦 ~ (ρ̅)1.7−2.3 , with the lower exponents

corresponding to topologies with larger characteristic feature size to domain size ratios. We

excluded the 20% dense samples from the scaling calculations, as the 20% dense spinodal

topologies displayed isolated islands on the inside of the models that depressed the mechanical

properties and unrealistically steepened the slopes. While these large exponents are not unexpected

at such large relative densities, this likely indicates that the solid spinodal topologies are bending

dominated and the spinodal arrangement of matter is not very mechanically efficient. The

numerical simulations indicate that solid spinodal models are yielding-limited (as opposed to

elastic buckling-limited) for all relative densities of interest, 20-70%.

23

Figure 11 – Numerical predictions of (a) Relative Young’s modulus and (b) relative yield strength

of spinodal solid models, as a function of their relative density, �̅�. Hollow markers indicate models

with isolated internal islands, which have been excluded from the scaling calculations. The

constituent material has Young’s modulus, 𝐸𝑠 = 210 𝐺𝑃𝑎 and yield strength, 𝜎𝑦𝑠 = 235 𝑀𝑃𝑎.

Different curves refer to models with different characteristic feature size, 𝜆. L denotes the domain

size.

Remarkably, these conclusions change dramatically for shell spinodal topologies (Fig. 12). Here,

𝐸 ~𝜎𝑦~ (ρ̅)1.2−1.3, a scaling that is very similar to that of the best stretching dominated lattice

materials. We attribute this exceptional performance to the deformation behavior of doubly curved

surfaces, which cannot be readily bent without the introduction of significant membrane stresses

24

– and hence deform in a stretching-dominated manner. For the base material yield strain used in

the calculation, shell spinodal topologies remain yielding-limited down to relative densities in the

order of 0.1%. Once the failure mechanism switches to buckling, the scaling worsens somewhat

(as expected), but remains reasonably close to one. Interestingly, the relationship between scaling

law exponent and characteristic feature size to domain size ratio, 𝜆/𝐿, is inverted compared to the

case of solid spinodal topologies. In shell spinodal topologies, a larger 𝜆/𝐿 corresponds to a

steeper scaling exponent for both stiffness and strength.

Figure 12 – Numerical predictions of (a) Relative Young’s modulus and (b) relative yield strength

of spinodal shell models, as a function of their relative density, �̅�. The constituent material has

Young’s modulus, 𝐸𝑠 = 210 𝐺𝑃𝑎 and yield strength, 𝜎𝑦𝑠 = 235 𝑀𝑃𝑎. Different curves refer to

25

models with different characteristic feature size, 𝜆. L denotes the domain size. The black dots mark

the transition from yielding to buckling failure.

The exceptional mechanical efficiency of shell spinodal topologies is attributed to its uniform

surface curvature (see Sec. 2.2), which results in a very uniform local stress distribution upon

loading, avoiding areas of substantial stress intensification. To confirm this hypothesis, we contrast

the local stress distribution upon uniaxial loading of a shell spinodal sample and a hollow

microlattice material of the same relative densities, reproduced from (Valdevit et al., 2013) (Fig.

13); the difference in Mises stress uniformity is clearly evident. In other words, shell spinodal

topologies are extremely efficient in terms of material utilization, with the entire structure yielding

(or buckling) nearly simultaneously, rather than localizing failure at the nodes (a well-known

shortcoming of hollow lattice materials (Meza et al., 2015, 2014; Schaedler et al., 2011; Torrents

et al., 2012; Valdevit et al., 2013).

Figure 13 – Comparison of von Mises stress distribution in a hollow microlattice unit cell with

bar aspect ratio ~6, and angle 𝜃 = 60° (Valdevit et al., 2013) and a spinodal shell model with

26

characteristic feature size, 𝜆

𝐿=

1

3, both at relative density of 0.01%, and under a compressive strain

of 0.1%. The constituent material has Young’s modulus, 𝐸𝑠 = 210 𝐺𝑃𝑎 and yield strength, 𝜎𝑦𝑠 =

235 𝑀𝑃𝑎. The stress limits (used to distinguish stress contours where 𝜎𝑣𝑚

𝜎𝑙𝑖𝑚𝑖𝑡≥ 1) are colored

“red”, with 𝜎𝑙𝑖𝑚𝑖𝑡 = 4 𝑀𝑃𝑎. Notice that the spinodal shell topology has a much more uniform

stress distribution, yielding excellent mechanical efficiency.

An obvious question emerges regarding the role of near-zero mean curvature on the uniformity of

the stress distribution and hence the mechanical efficiency of shell spinodal topologies. As clearly

illustrated in Fig. 6, all 50%-derived spinodal shell topologies possess the characteristics of

minimal surfaces (i.e., κ1 = −κ2 almost everywhere on the surface), while the same is not true for

shell spinodal topologies derived from solid spinodals with densities different from 50%. To

quantitatively ascertain the impact of zero mean curvature on the mechanical properties of spinodal

shell topologies, we generated finite element models of 20% and 70%-derived spinodal shell

topologies at two different characteristic feature sizes ( λ

L =

1

8 and

λ

L =

1

3 ), and compared their stress-

strain curve to those of 50%-derived topologies. All shell thicknesses were chosen so that all shell

topologies had a 10% relative density. The results are reported in Fig. 14. Two interesting results

emerge: (i) 50%-derived shell spinodal topologies outperform both the 20% and the 70%-derived

topologies, at both characteristic length scales, confirming that the minimal surface characteristics

of 50%-derived spinodal structures are partly responsible for their mechanical efficiency. This is

attributed to a more homogeneous stress distribution (see insets). (ii) The mechanical performance

of topologies derived from solid spinodal structures with densities significantly different from 50%

seem to be largely unaffected by the characteristic feature size, in contrast with 50%-derived

spinodal topologies.

27

Figure 14. The effect of mean surface curvature, H, on the mechanical efficiency of shell spinodal

topologies. The insets display contours of von Mises stress. Notice that spinodal shell topologies

derived from 50%-dense spinodal solids (and hence exhibiting near-zero mean curvature

everywhere on the surface) are more efficient than spinodal shell topologies of equal relative

density but derived from 20% and 70% solid spinodals (and hence having non-zero mean

curvatures throughout the surface). The constituent material has Young’s modulus, 𝐸𝑠 = 210 𝐺𝑃𝑎 and yield strength, 𝜎𝑦𝑠 = 235 𝑀𝑃𝑎 . The stress limits (used to distinguish stress

contours where 𝜎𝑣𝑚

𝜎𝑙𝑖𝑚𝑖𝑡≥ 1 ) are colored “red”, with (a) 𝜎𝑙𝑖𝑚𝑖𝑡 = 2.2 𝑀𝑃𝑎 and (b) 𝜎𝑙𝑖𝑚𝑖𝑡 =

2 𝑀𝑃𝑎.

It is instructive to compare the mechanical efficiency of solid and shell spinodal topologies to that

of a wide range of strut-based and shell-based architected materials, in particular solid strut

metallic nanolattices (Khaderi et al., 2017), solid strut carbon nanolattices (Bauer et al., 2016b),

hollow strut metallic microlattices (Schaedler et al., 2011; Zheng et al., 2014), hollow strut ceramic

nanolattices (Meza et al., 2015, 2014), and two mechanically efficient metallic shell-based

architected materials, the P-surface material (Han et al., 2015) and the D-surface material (Han et

al., 2017) . These comparisons are presented in Fig. 15 in terms of relative Young’s modulus,

28

𝐸/𝐸𝑠, and relative yield strength, 𝜎𝑦/𝜎𝑦𝑠, versus relative density, �̅�. Only spinodal topologies with

𝜆/𝐿 = 1/5 are depicted for clarity, for two different values of the base material yield strain:

휀𝑦𝑠=0.01 (representative of IP-Dip, thus allowing direct comparison with experimental results) and

휀𝑦𝑠=0.001 (representative of a metal). As all the results depicted in Fig. 15 for the 10 comparison

structures are obtained experimentally, experimental results for solid and shell spinodal topologies

(obtained as described in Sec. 3.2) are depicted as well. Also plotted on Fig. 15 are the Hashin-

Shtrikman upper bound for stiffness (Hashin and Shtrikman, 1963; Khaderi et al., 2017) and the

Suquet-Ponte-Castaneda nonlinear upper bound for strength (Castaneda and Debotton, 1992;

Suquet, 1993), which represent theoretical limits for the mechanical performance of isotropic

cellular materials. We emphasize that solid and shell spinodal architectures are the only isotropic

topologies in Fig. 15. Notice that while spinodal solid topologies are worse than double gyroid

materials and glassy carbon lattices at high relative densities, shell spinodal topologies are on par

with or better than all referenced ultralight topologies across a relative density range covering 3

orders of magnitude, clearly demonstrating their exceptional mechanical efficiency.

While the model predictions for spinodal shell topologies at ultra-low density (i.e., in the buckling-

limited regime) are not verified experimentally due to fabrication challenges, the remarkable

imperfection insensitivity of spinodal shell topologies demonstrated in Sec. 3.2 strongly suggest

that very small knock-down factors will apply, providing confidence in the numerical results.

29

Figure 15 – Comparison of the mechanical performance of solid and shell spinodal topologies

and well-established strut-based and shell-based architected materials: hollow octahedral-type

metallic microlattices (Schaedler et al., 2011), hollow octet ceramic nanolattices (Meza et al.,

2014), hollow octet metallic microllatices (Zheng et al., 2014), hollow octahedron-octet

nanolattices (Meza et al., 2015), solid glassy carbon nanolattices (Bauer et al., 2016b), double

gyroid nanolattices (Khaderi et al., 2017), P-surface (Han et al., 2015) and D-surface (Han et al.,

2017) architected materials.

Only spinodal topologies with characteristic feature size, 𝜆/𝐿 = 1/5 are depicted for clarity, for

two different values of the base material yield strain: 휀𝑦𝑠=0.01 (representative of IP-Dip, thus

allowing direct comparison with experimental results) and 휀𝑦𝑠=0.001 (representative of a metal).

Also depicted are the Hashin-Shtrikman upper bound for stiffness (Hashin and Shtrikman, 1963;

Khaderi et al., 2017) and the Suquet-Ponte-Castaneda nonlinear upper bound for strength

(Castaneda and Debotton, 1992; Suquet, 1993), which represent theoretical limits for the

mechanical performance of isotropic cellular materials.

4. Conclusions

The mechanical performance of spinodal topologies was numerically investigated and

experimentally verified. Cellular materials with spinodal architectures were numerically generated

by solving the Cahn-Hillard evolution equation with a Finite Difference numerical scheme.

Topologies were extracted at different evolution times, corresponding to different characteristic

feature size to domain size ratios, and different relative densities, i.e., different ratios of solid to

void phase. We refer to these topologies as solid spinodal models. Another class of spinodal

30

topologies was generated by retaining the interface between the solid and void phases of solid

spinodal models and eliminating both volumetric phases: we refer to these topologies as shell

spinodal models. The characteristic feature size of spinodal models was calculated by Fast Fourier

Transform analysis, and used as a metric for ‘unit cell size’ of these stochastic cellular materials.

This feature size was shown to scale with the cubic root of the evolution time, in agreement with

the theory of spinodal decomposition. The distribution of principal curvatures of all spinodal

models was also extracted, and verified to be representative of naturally occurring spinodal

systems. The mechanical performance of solid and shell spinodal topologies was investigated

numerically by finite element analysis. The numerical models, verified by mechanical experiments

performed on solid and shell spinodal samples fabricated at the microscale by two-photon

polymerization Direct Laser Writing (DLW), showed three key results: (i) shell spinodal

topologies are remarkably efficient, displaying stretching dominated behavior over a very wide

range of relative densities; (ii) whereas shell-based architected materials are generally imperfection

sensitive, shell spinodal topologies display remarkable imperfection insensitivity, which is

expected to result in superior mechanical response at ultra-low relative densities, where buckling

knock-down effects plague the response of most architected materials; (iii) in the low-relative

density regime (�̅� < 1%) architected materials with shell spinodal topologies outperform most

strut and shell-based architected materials in terms of specific stiffness and strength, while

displaying complete isotropy in three dimensions.

These remarkable properties are entirely due to the topological characteristics of the architecture,

and are attributed to the very uniform double curvature across the entire surface of the material,

which avoids stress intensifications and prevents local bending deformations of the surface without

the introduction of membrane stresses, thus imparting stretching dominated behavior to the cellular

31

material. Importantly, base material effects have not been considered in this analysis: if emerging

nano-scale fabrication techniques are used to exploit size effects in metallic and ceramic base

materials (Khaderi et al., 2017; Mallory and Hajdu, 1990; Schwartzberg and Olynick, 2015),

unprecedented specific strength can be achieved. These opportunities are currently under

investigation.

Finally, we emphasize that the true potential of these topologies stems from the fact that they can

be self-assembled, using a variety of physical processes at multiple length scales. While the

experiments in this article are performed on 3D printed samples to facilitate fabrication of samples

of controlled size and shape and application of mechanical loads, macroscale samples can be

produced by selective etching of a metallic solid solution (Hodge et al., 2006) or by phase

separation of a colloidal suspension (bijel technology) (Lee and Mohraz, 2010), resulting in

nanoscale and microscale characteristic feature size, respectively. These self-assembly

methodologies can be combined with additive manufacturing at a larger scale to create macroscale

architected materials with unprecedented structural hierarchy, vastly improving the scalability of

micro/nano-architected materials processing.

Acknowledgments

This work was supported by the Air Force Office of Scientific Research (AFOSR), Contract #

FA9550-14-1-0352 (PM: Dr. Joycelyn Harrison). J.B. gratefully acknowledges financial support

from the Deutsche Forschungsgemeinschaft (DFG), grant BA 5778/1-1. The ABAQUS Finite

Element Analysis software is licensed from Dassault Systemes SIMULIA, as part of a Strategic

Academic Customer Program between UC Irvine and SIMULIA. SEM imaging and in-situ

mechanical testing were conducted at the Irvine Materials Research Institute (IMRI).The authors

32

are thankful to Prof. Ali Mohraz for useful discussions and to S.W. Godfrey for coding the

parallelized finite difference algorithm for solution of the Cahn-Hillard equation.

Appendix 1 – Synopsis of algorithm for spinodal decomposition

The spinodal decomposition equation (Eq. (1)) is solved numerically with a Finite Difference

scheme, as described in detail in Sun et al., 2013. Here we report the key details of the algorithm

for completeness.

Equation (1) is solved on a cubic volume with edge length, L=100, which is discretized into a

lattice of mesh size, ℓ = 𝐿/100 = 1. Let 𝑢𝑖𝑗𝑘𝑚 denote the discrete value of the phase field variable

𝑢(𝑖, 𝑗, 𝑘, 𝑚𝜏) at nodal point (𝑖, 𝑗, 𝑘), where 𝜏 is the integration time step (𝜏 =0.005 is chosen as a

good compromise between solution accuracy and computational cost, as suggested by (Sun et al.,

2013) and m is the number of time steps.

A finite difference scheme is used to discretize equation (1) as:

𝑢𝑖𝑗𝑘𝑚+1−𝑢𝑖𝑗𝑘

𝑚

𝜏= ∆[ (𝑢𝑖𝑗𝑘

𝑚 )3

− 𝑢𝑖𝑗𝑘𝑚 − 𝜃2∆𝑢𝑖𝑗𝑘

𝑚 ] (A1)

where again 𝜃 represents the thickness of the phase A / phase B interface (here chosen as 1.1) and

∆ is the Laplacian operator. Equation (2) is solved with periodic boundary conditions:

𝑢(𝑖, 𝑗, 𝑘, 𝑚𝜏) = 𝑢(𝑖 + 𝐿, 𝑗, 𝑘, 𝑚𝜏) (A2.1)

𝑢(𝑖, 𝑗, 𝑘, 𝑚𝜏) = 𝑢(𝑖, 𝑗 + 𝐿, 𝑘, 𝑚𝜏) (A2.2)

𝑢(𝑖, 𝑗, 𝑘, 𝑚𝜏) = 𝑢(𝑖, 𝑗, 𝑘 + 𝐿, 𝑚𝜏) (A2.3)

and a randomly generated initial condition, 𝑢(𝑖, 𝑗, 𝑘, 0) = 𝑢0(𝑖, 𝑗, 𝑘, 0) ∈ [−5,5] ∗ 10−4 ≠ 0.

Here, phase A (𝑢 = −1) represents void space, and phase B (𝑢 = 1) represents solid material. As

the solution progresses, the system phase separates at relative early times, and subsequently

33

continues to coarsen; during the coarsening phase, the curvature of the interface between solid and

void decreases and the size of the single-phase domains increases.

To enforce a specific relative density (i.e., a prescribed ratio of phase A and phase B regions), the

threshold method with the cutoff 𝑢𝑐𝑚 is used. The phase value of point (𝑖, 𝑗, 𝑘) at time 𝑚𝜏 is

defined as:

𝑃𝑉𝑖𝑗𝑘𝑚 = 𝐻(𝑢𝑖𝑗𝑘

𝑚 − 𝑢𝑐𝑚) (A3)

where 𝐻(𝑢𝑖𝑗𝑘𝑚 − 𝑢𝑐

𝑚) is the Heaviside function. When 𝑢𝑖𝑗𝑘𝑚 − 𝑢𝑐

𝑚 > 0, then 𝑃𝑉𝑖𝑗𝑘𝑚 = 1, indicating

the space is occupied by material. When 𝑢𝑖𝑗𝑘𝑚 − 𝑢𝑐

𝑚 < 0 then 𝑃𝑉𝑖𝑗𝑘𝑚 = 0 indicating void space. For

a desired relative density �̅�, the cutoff 𝑢𝑐𝑚 is adjusted such that:

�̅� =1

𝐿3∑ ∑ ∑ 𝑃𝑉𝑖𝑗𝑘

𝑚𝐿1

𝐿1

𝐿1 =

1

𝐿3∑ ∑ ∑ 𝐻(𝑢𝑖𝑗𝑘

𝑚 − 𝑢𝑐𝑚)𝐿

1𝐿1

𝐿1 (A4)

With this approach, dense spinodal topologies with arbitrary relative densities are generated.

Appendix 2 – Algorithm for extraction of curvature probability maps

The principal curvatures of a surface at a point P, 𝜅1 and 𝜅2, are defined as the minimum and

maximum curvatures at that point, respectively. A paraboloid fitting method can be used to extract

these curvatures at each point of the shell spinodal meshes, as proposed in (Gatzke and Grimm,

2006; Surazhsky et al., 2003). The following procedure is implemented: (i) The triangular shell

mesh (generated as described in sec. 2.1) is imported in Matlab. (ii) For any node P in the surface

mesh, a number J of elements is identified as the set of elements belonging to the three neighboring

rings of node P, with the first neighboring ring defined as all the elements sharing node P, and the

next two rings as the elements sharing the boundary nodes of the previous ring. (iii) Unit normal

vectors to each element in the set J is are constructed by normalizing the cross product of two

34

edges of the element. (iv) The unit normal of the surface at node P, np, is obtained as the average

of the unit normals of all the elements in the first three neighboring rings of the point P as follows

𝐧P =1

J∑ 𝐧e

J1 (A5)

where ne is the unit normal of element “e”. (v) All the nodes in the first three rings of P, along

with P, are transformed using a rotation matrix such that 𝐧p align with the z-axis of the global

coordinate system. (vi) The transformed coordinates of these nodes are used to least-square fit the

coefficients ao to a5 of the biquadratic polynomial surface:

zv = ao + a1x + a2y + a3xy + a4x2 + a5y2 (A6)

(vii) The Hessian matrix of the polynomial surface is calculated as:

[2𝑎4 𝑎3

𝑎3 2𝑎5] (A7)

(viii) The pairs of minimum and maximum principal curvatures, 𝜅1 and 𝜅2, at P are extracted as

the eigenvalues of the Hessian matrix. (ix) The process above is repeated to find the principal

curvatures at every single node of the spinodal shell mesh. Outliers in 𝜅1 and 𝜅2 are determined

based on a median absolute deviation method (Leys et al., 2013), and any (𝜅1, 𝜅2) pair that has an

outlier is removed. (x) Principal curvature density function maps are generated as contour plots of

the probability density function 𝑃(𝜅1, 𝜅2), defined as:

𝑃(𝜅1, 𝜅2) = 𝑁(𝜅1,𝜅2)

∑ 𝑁(𝜅1,𝜅2) (A8)

where 𝑁(𝜅1, 𝜅2) is the number of (𝜅1, 𝜅2) pairs having curvatures in the range [(𝜅1, 𝜅2), (𝜅1 +

∆𝜅1, 𝜅2 + ∆𝜅2)]. Here, 𝛥𝜅1 = max(𝜅1)−min (𝜅1)

𝑁1 and 𝛥𝜅2 =

max(𝜅2)−min (𝜅2)

𝑁2 , with 𝑁1 and 𝑁2 the

number of intervals in 𝜅1 and 𝜅2, respectively.

This procedure was used to generate the contour plots in Fig. 6.

35

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Supplementary Material

Fig. S1. SEM images of spinodal samples fabricated in IP-Dip by two-photon polymerization

Direct Laser Writing. All samples have characteristic feature size, 𝜆

𝐿=

1

5, with L the domain size.

(a) Spinodal solid model, �̅� = 50%. (b) Spinodal solid model, �̅� = 30%. (c) Spinodal solid model,

�̅� = 20%. (d) Spinodal shell model, �̅� = 10%. (e) Spinodal shell model, �̅� = 7%. (f) Spinodal

shell model, �̅� = 5%.

43

Fig. S2. Uniaxial response of the material IP-Dip, measured by compression experiments on

micro-pillars of aspect ratio of 4 and different diameters, manufactured by two-photon

polymerization Direct Laser Writing with a Nanoscribe Photonics GT Professional.